We use the term utility in its broad sense in cardinal welfare economics as a “measurement of the higher-order characteristic that is relevant to the particular distributive justice problem at hand” [1]}. Similarly to [2]}, [3]}, [4]}, we define the utility of a user as the ranking performance, and the utility of an item as its average exposure to users, which are formalized in (REF ) below. Utilities are defined according to the position-based model [5]}, [6]}, [4]} with weights \({b}\in \mathbb {R}_+^m\) . The weight \({b}_k\)  is the probability that a user examines the item at position \(k\) , and we assume that the weights are non-increasing. Since there are \(K\)  recommendation slots, we have \({b}_1 \ge \ldots \ge {b}_K\)  and \({b}_k = 0\)  for any \(k > K\) . The user and item utilities are then:
\(\hspace{-15.0pt}\leavevmode \xbox {resizebox}{\XMLaddatt {width}{392.83922pt}\displaystyle \textit {User utility: }\, u_i(P) = \sum _{j=1}^m \mu _{ij}P_{ij}^\top {b}\quad \quad \textit {Item exposure: }\, v_j(P) = \sum _{i=1}^nP_{ij}^\top {b}.}\) 
